Let $\mathcal{U} = \{U_i\}_{i\in I} $ be a collection of open sets with the property that the set $\bigcap_{i\in J} U_i $ is open for all subsets $J$ of $I$.
Is there a name for such collections of open sets?
Both locally finite collections and point-finite collections have this property, but these notions are too strong (just think of infinite discrete spaces).
Such collections have been called interior preserving, as in the definition of orthocompact space found here. An older and less descriptive term is Q-collection, as in this paper.